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Evaluate \int \frac{x}{\left ( x-3 \right )\sqrt{x+1}}dx

  • Option 1)

    -2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

  • Option 2)

    -2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

  • Option 3)

    2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

  • Option 4)

    2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

 

Answers (1)

best_answer

As we learnt

Case for special type of indefinite integration -

\int x^{m}(a+bx^{n})^{p}dx

When P is an integer if P> 0 then apply expanded form

P< 0 then we put x=t^{k}

- wherein

Where k is the common denominator of m and n

 

 

Put x + 1 = t2. We get

                        I=\int \frac{\left ( t^{2}-1 \right )2tdt}{\left ( t^{2}-4 \right )t}=2\int \frac{t^{2}-1}{t^{2}-4 }dt

                        =2\int \left \{ 1+\frac{3}{t^{2}-4} \right \}dt=2t+\frac{3}{2}\ln \left | \frac{t-2}{t+2} \right |+c

            =2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c.

 


Option 1)

-2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

Option 2)

-2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

Option 3)

2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

Option 4)

2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c

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gaurav

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